Global Positivity Conditions

In this chapter, we study some global positivity conditions, notably ampleness, nefness, bigness, and pseudo-effectivity. For these conditions, we require not only the relative (vertical) positivity but also a lower bound for the height function (horizontal positivity). We show that the horizontal positivity can also be expressed as a lower bound of the asymptotic minimal slope. This is analogous to the condition of base locus freeness in algebraic geometry. In Sects. 9.1 and 9.2, we define these positivity conditions and discuss the links with the corresponding relative positivity conditions and the positivity of asymptotic sectional invariants. Section 9.3 is a reminder on the construction of the canonical metric family of an arithmetic dynamical system. Then, in Sect. 9.4, we prove a theorem of Bogomolov conjecture type in the framework of Abelian varieties over an adelic curve with Archimedean places. In Sect. 9.5, we discuss arithmetic dynamical systems.